We will be interested in (what we will call generically tame) families \(\mathcal {A}\)  of universally Baire sets which are
constructibly closed (i.e. \(\mathcal {P}\left(2^\omega \right)^{L(\mathcal {A})}=\mathcal {A}\) ), countably closed (e.g.
\(\mathcal {A}^\omega \subseteq L(\mathcal {A})\) ), and generically invariant (e.g. the theory of \(L(\mathcal {A})\)  cannot be changed by set sized forcing). One example will be given by
\(\mathcal {A}=\mathcal {P}\left(2^\omega \right)^{L(\mathrm {Ord}^\omega )}\)  assuming large cardinalsThis remarkable result of Woodin is to my knowledge unpublished. There are some handwritten notes of a proof sketch by Larson, and it is mentioned in [1]}., another by the class \(\text{{\sf UB}}\)  itself as computed in any generic extension of \(V\)  collapsing a supercompact to countable.
